1. What Is the Complement of a Set?
The complement of a set shows everything that is not in the set, but still belongs to the universal set. It helps describe the “outside” part of a set.
If A is a set inside a universal set U, then the complement of A contains all the elements of U that are not in A.
2. Notation
The complement of A is written as:
- A′ (A prime)
- Sometimes as Ac
Both represent the same idea.
3. Meaning of A′
A′ contains:
- Every element in the universal set U
- Except the elements of A
3.1. Symbolic Form
\( A' = U - A \)
4. Examples
Here are some simple examples to understand complements:
4.1. Example 1
If:
U = \{1,2,3,4,5\}, \quad A = \{2,3\}
Then:
A′ = \{1,4,5\}
4.2. Example 2
If U is the set of all lowercase letters:
A = \{a,e,i,o,u\}
Then A′ contains all lowercase consonants.
4.3. Example 3 (Empty Set Case)
If A is empty:
A = \emptyset \Rightarrow A' = U
5. Complement in Venn Diagrams
In a Venn diagram, the complement of A is shown by shading the region outside circle A, but still inside the universal rectangle U.
6. Double Complement
If you take the complement twice, you get the original set back.
6.1. Formula
\( (A')' = A \)
7. Important Points
A few key things to remember about complements:
7.1. Key Ideas
- The complement always depends on the universal set U.
- A and A′ never overlap; they are completely separate.
- A ∪ A′ = U (together they fill the universal set).
- A ∩ A′ = \emptyset (no element can be in both).